ABSTRACT:
We aim to indicate how the principles of completeness and iteration
can be used to show that the modern theory of large cardinals arises naturally
from basic considerations in the theory of forcing. Goedel's L, the smallest
inner model of ZFC, is incomplete in the sense that by forcing over L one
can witness new absolute properties. We show that to resolve this kind of
incompleteness, one is necessarily led to the existence of 0#, the
smallest "large cardinal". Then we show that inner models for larger
cardinals can be obtained from L by iterating the "#-operation". Moreover, the
incompleteness of these models can only be resolved by assuming the
consistency of very strong large cardinal axioms. In this sense the
principles of completeness and iteration necessarily lead to measurable
cardinals of high order, and we conjecture that they in fact lead to us to
the existence of inner models containing superstrong cardinals.